Find the volume and surface area of a cube.
Volume is the side length cubed; surface area is six times one face's area, since a cube has six equal faces.
Volume: side³
Surface area: 6 × side²
A cube is a box whose length, breadth and height are all equal, like a dice or a Rubik's cube. The cube volume calculator takes the side length and gives two things: the volume, side³, which is how much space the cube holds, and the total surface area, 6 × side², which is how much material covers its outside.
Volume matters when filling or storing: how many litres a cubical water tank holds, or how much concrete a cubical foundation block needs. Surface area matters when covering or making: the cardboard for a gift box, the paint for a block or the tiles for a cubic planter. The two grow at different rates, which is why the calculator shows both together. Cubes are part of Class 8 and Class 9 mensuration.
1. Measure one edge of the cube, a.
2. Volume = a × a × a = a³.
3. Area of one face = a².
4. Total surface area = 6 × a², because a cube has six identical square faces.
5. Lateral surface area, if needed, is 4 × a², leaving out the top and bottom.
6. To convert a volume in cm³ to litres, divide by 1,000. For m³, multiply by 1,000.
A cube of side 4 can be filled with a 4 × 4 layer of 16 unit cubes, and it takes 4 such layers to fill it, giving 64 unit cubes. That is 4³. The same reasoning gives l × b × h for any cuboid, and the cube is the case where all three are equal. This is also where the word cubed comes from, just as squared comes from the area of a square.
If the side doubles, the surface area becomes 4 times as large but the volume becomes 8 times as large. The ratio of surface to volume, 6a² ÷ a³ = 6 ÷ a, falls as the cube gets bigger. This square-cube law explains why large packages use less cardboard per litre of contents, why big animals lose heat more slowly than small ones, and why crushed sugar dissolves faster than a sugar cube: smaller pieces expose more surface.
Volume is in cubic units, so conversions use the cube of the linear factor. One metre is 100 cm, but 1 m³ is 10,00,000 cm³, which is 1,000 litres. A cube's face diagonal is a√2, and its space diagonal, from one corner to the opposite corner through the middle, is a√3. That gives the longest rod that fits inside. The calculator reports volume and total surface area only; the diagonals can be worked out from the same side.
The Sharma family is building a cubical overhead water tank with an inner side of 1.5 metres and wants its capacity and the area to be waterproofed.
Answer: Volume 3.375; Surface area 13.5
Multiplying the side by 3 instead of cubing it. A 5 cm cube is 125 cm³, not 15 cm³.
Using outer dimensions to find the capacity of a tank with thick walls. Use the inner side.
Converting cm³ to m³ by dividing by 100 instead of by 10,00,000.
Counting 4 faces for total surface area, or 6 faces when an open-top box needs only 5.
Reporting volume in square units or surface area in cubic units.
Finding the capacity of cubical water tanks and storage bins in litres.
Estimating concrete for cubical footings or test cubes used in construction.
Working out cardboard or wrapping paper for cube-shaped gift boxes.
Packing problems, such as how many small cubes fit in a larger one.
Class 8 to 10 mensuration and aptitude problems on cubes.
Why six faces for surface area?
A cube has exactly six square faces, all the same size, so the total surface area is six times the area of just one face.